Periods in partial words: An algorithm

Authors:
F. Blanchet-Sadri;Travis Mandel;Gautam Sisodia
Affiliations:
Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402-6170, USA;Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, USA;Department of Mathematics, University of Washington, P.O. Box 354350, Seattle, WA 98195-4350, USA
Venue:
Journal of Discrete Algorithms
Year:
2012

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Abstract

Partial words are finite sequences over a finite alphabet that may contain some holes. A variant of the celebrated Fine-Wilf theorem shows the existence of a bound L=L(h,p,q) such that if a partial word of length at least L with h holes has periods p and q, then it also has period gcd(p,q). In this paper, we associate a graph with each p- and q-periodic word, and study two types of vertex connectivity on such a graph: modified degree connectivity and r-set connectivity where r=qmodp. As a result, we give an algorithm for computing L(h,p,q) in the general case and show how to use it to derive the closed formulas.